RESEARCH Revista Mexicana de Fısica59 (2013) 453–459 SEPTEMBER-OCTOBER 2013

The band gap problem: the accuracy of the Wien2k code confronted

J. A. Camargo-Martınez and R. BaqueroDepartamento de Fısica, CINVESTAV-IPN,

Av. IPN 2508, 07360 Mexico.

Received 1 February 2013; accepted 26 April 2013

This paper is a continuation of our detailed study [Phys. Rev. B 86, 195106 (2012)] of the performance of the recently proposed modifiedBecke-Johnson potential (mBJLDA) within the known Wien2k code. From the 41 semiconductors that we have considered in our previouspaper to compute the band gap value, we selected 27 for which we found low temperature experimental data in order to pinpoint the relativesituation of the newly proposed Wien2k(mBJLDA) method as compared to other methods in the literature. We found that the GWA gives themost accurate predictions. The Wien2k (mBJLDA) code is slightly less precise, in general. The Hybrid functionals are less accurate, on theoverall. The GWA is definitely the most precise existing method nowadays. In 88% of the semiconductors considered the error was less than10%. Both, the GWA and the mBJLDA potential, reproduce the band gap of 15 of the 27 semiconductors considered with a 5% error or less.An extra factor to be taken into account is the computational cost. If one would seek for precision without taking this factor into account,the GWA is the method to use. If one would prefer to sacrifice a little the precision obtained against the savings in computational cost, theempirical mBJLDA potential seems to be the appropriate method. We include a graph that compares directly the performance of the bestthree methods, according to our analysis, for each of the 27 semiconductors studied. The situation is encouraging but the problem is not yeta closed issue.

Keywords: Band gap problem; Wien2k; mBJLDA potential; hybrid functionals; GW approximation.

PACS: 71.15.Mb; 71 .20.Mq; 71. 20.Nr

1. Introduction

Khon-Sham equations [1] are central to the practical appli-cation of Density Functional Theory (DFT). To solve them,an approximation to the exchange and correlation energy isrequired from which an exchange and correlation potential isderived. The way in which this term is approximated is cru-cial to the proper description of the band structure of solids.The Local Density Approximation (LDA) [2], the General-ized Gradient Approximation (GGA) [3–5] and the meta-GGA [3, 6], among others, describe very well the electronicband structure of even complicated metallic systems. Theyfail, nevertheless to account for the band gap value of semi-conducting systems, a short come known for several yearsnow [7]. Efforts to solve this problem were done since longago. Approximations as the “scissor operator” [8], the Lo-cal Spin Density Approximation, LSDA+U [9] and methodsbased on the use of Green’s functions and perturbation the-ory as the GW approximation, GWA [10–12], were proposed.In the last ten years, these efforts gave rise to substantiallyimproved results. Some of the new proposals include, thescreened hybrid functional of Heyd, Scuseria and Ernzerhof(HSE) [13–15] and the middle-range exchange and correla-tion hybrid functional of Henderson, Izmaylov, Scuseria andSavin (HISS) [16, 17]. Another recent proposal is the mod-ified Becke-Johnson potential (mBJLDA) proposed by Tranand Blaha [18]. This potential was introduced to the Wien2kcode [19] in 2010.

2. The mBJLDA potential

Recently, we made a detailed analysis of the mBJLDA poten-tial based on the calculation of the electronic band structure

of 41 semiconductors [20]. This paper is a continuation ofthat work. We found an important improvement in the pre-dictions of the band gap as compared to experiment. ThemBJLDA potential [18] is a empirical potential of the form

V MBJx,σ (r) = cV BR

x,σ(r) + (3c− 2)1π

√512

√2tσ(r)ρσ(r)

(1)

whereρσ(r) is the spin dependent density of states,tσ(r) isthe kinetic energy density of the particles with spinσ. andV BJ

x,σ (r) is the Becke-Roussel potential (BR) [21]. The cstands for

c = α + β

(1

Vcell

∫d3r

| ∇ρ(r) |ρ(r)

)1/2

(2)

α and β are free parameters. The Wien2k code definesα = −0.012 and β = 1.023 Bohr1/2. These values aregeneral but certainly fixed experimenting with several cases.A particular feature of this potential is that a correspond-ing exchange and correlation energy term ,Exc[ρ], such thatthe mBJLDA potential is obtained in the usual way, namely,Vxc = δExc[ρ]/δρ, is not possible. As a consequence, aconsistent optimization procedure to obtain the lattice pa-rameters, the Bulk modulus and its derivative with respectto pressure are not actually possible. This is a consequenceof the empirical character of this potential. For that rea-son, Tran and Blaha have proposed the empirical alternativethat prior to a band structure calculation with the mBJLDApotential, the lattice parameter is found from either a LDAor a GGA optimization procedure and the result introducedinto the code to perform the band structure calculation ofthe semiconductor system. Such a procedure gives rise toquite improved results as compared to the previous version of

454 J. A. CAMARGO-MARTINEZ AND R. BAQUERO

the Wien2k code. It is known that the LDA underestimatesas a rule, the lattice parameters and, on the contrary, GGAoverestimates them. We have explored the possibility of us-ing the averaged value as the lattice parameter,aAvg, whereaAvg = (aLDA + aGGA)/2. HereaLDA (aGGA) is the latticeparameter obtained from an LDA (GGA) optimization pro-cedure. WhenaAvg is used as input into the Wien2k code im-plemented with the mBJLDA potential, a better agreement ofthe band gap value with experiment is obtained as comparedto the results with eitheraLDA or aGGA. So this procedureturns out to give better results than the one recommended byTran and Blaha and its extra computational cost is relativelylow. A surprising result was, nevertheless, obtained whenthe experimental low temperature lattice parameter,aLT , wasintroduced instead. Unexpected deviations of the band gapvalue from experiment as big as 48% were obtained [20].This is a disturbing result since the lattice parameters ob-tained from any optimization procedure are judged to be asgood as the deviation from the experimental lattice param-eter value is small, and so one expects to get the best result(the minimum deviation of the predicted band gap value fromexperiment) when the experimental lattice parameter is used.This is not the case. This fact throws doubts on the meaningof the optimization procedure altogether when the empiricalmBJLDA code is employed. Nevertheless, we stress that theresults obtained for the band gap value of semiconductors us-ing the mBJLDA potential represents a relevant improvementat relatively low computational cost, a fact that we will em-phasize below.

3. The HSE method

Hybrid functionals are a linear combination of Hartree-Fock(HF), LDA and GGA terms and were proposed initially withthe aim of improving LDA and GGA in the calculation ofthe energy bands of molecules [22, 23]. More recently, hy-brid functionals were used as an effort to improve the old-standing problem of the band gap of semiconductors; theyinclude the Heyd-Scuseria-Ernzerhof (HSE) functional [13]proposed in 2003. It combines a screened short-range HFterm and a screened short- and long-range functional pro-posed by Perdew, Burke and Ernzerhof (PBE) [4]. Thescreened terms in HSE result from splitting the Coulomb op-erator into short- and long-range terms in the following way

1r

=erfc(ωr)

r︸ ︷︷ ︸SR

+erf(ωr)

r︸ ︷︷ ︸LR

, (3)

where the complementary error function

erfc(ωr) = 1− erf(ωr)

andω determines the range. The functional form of HSE isbased on the hybrid functional of Perdew, Burke and Ernz-erhof (PBEh) [24] (also known in the literature as PBE1PBEand PBE0) [25,26], as follows

EPBEhxc = aEHF

x + (1− a)EPBEx + EPBE

x + EPBEc , (4)

The expression for the HSE exchange-correlation energy,EHSE

xc , is

EHSExc = aEHF,SR

x (ω) + (1− a)EωPBE,SRx (ω)

+ EωPBE,LRx (ω) + EPBE

c , (5)

whereEHF,SRx is the HF short-range functional (SR),EωPBE,SR

x

andEωPBE,LRx are the short-range (SR) and long-range (LR)

components of the PBE functional.a is a mixing constantthat is derived from perturbation theory [27]. In the litera-ture, the functional HSE appears as HSE03 and HSE06. Thedifference is in the choice of the value ofω. We will referto the HSE03 simply as HSE, in this work. In 2005, Heydetal. [14] reported a study of the band gap and lattice parame-ters of semiconductor compounds using the HSE functional.We will comment on these results below.

Recently, Marqueset al. [28] have proposed to relate themixing constanta to dielectric properties of the solid. Theytook a = 1/εPBE

∞ . Their calculation using the hybrid func-tional PBE0 improves the predictions for the band gap valueof the 21 semiconductors considered as compared to the orig-inal formulation. Furthermore, they useda ∼ g, anda ∼ g4

whereg takes the form of the term in parenthesis in Eq. (2).They introduced this form of thea parameter into the hy-brid functionals PBE0 and HSE06, respectively, and got animproved result. These proposals improve the performanceof the hybrid functionals at no extra cost. We will commentfurther on these results below.

4. The HISS potential

Another successful potential to calculate the band structureof semiconductors is the middle-range hybrid exchange andcorrelation Henderdon-Izmaylov-Scuderia-Savia functional(HISS) [16,17]. It also uses the PBE potential but in a differ-ent way,

EHISSxc =ESR-PBE

x +ELR-PBEx

+EPBEc +cMR(EMR-HF

x −EMR-PBEx ), (6)

where the last two terms in parentheses are the middle-range(MR) exact exchange and middle-range PBE exchange ener-gies, given by

1r

=erfc(ωSRr)

r︸ ︷︷ ︸SR

+erf(ωSRr)

r︸ ︷︷ ︸LR

+erf(ωSRr)− erf(ωLRr)

r︸ ︷︷ ︸MR

, (7)

In 2012, Luceroet al. [29] reported their study of theband gap and lattice parameters of some semiconductor com-pounds using HISS, withωSR = 0.84a−1

0 , ωLR = 0.20a−10 y

cMR = 0.60. These values were determined by fitting them tosome atomization energies, barriers hights and values of thegap for some compounds.

Rev. Mex. Fis.59 (2013) 453–459

THE BAND GAP PROBLEM: THE ACCURACY OF THE WIEN2K CODE CONFRONTED 455

5. The GW approximation (GWA)

As it is well known, the many-body Shrodinger equation con-tains the Coulomb interaction term which is a two-body po-tential and creates the difficulty to solve it for realistic sys-tems. To address this problem, the Hartree-Fock Approx-imation (HFA) adds to the average Coulomb potential (theHartree term) a non-local exchange potential which reflectsthe Pauli Exclusion principle. The energy gap of semicon-ductors predicted in this way turns out to be in most cases toolarge. This is due to the neglect of correlations or screeningwhich are crucial in solids. To simulate the effect of corre-lations, Slater introduced theXα approximation which maybe regarded as a precursor of the modern DFT. In DFT, theground state energy can be proved to be a functional of theground state density but the explicit form of the functionalis not known. The minimization of the total energy func-tional with respect to the density gives the Kohn-Sham equa-tions. The unknown exchange and correlation potential is ap-proximated either by the local density approximation (LDA),the generalized gradient approximation (GGA) or the meta-GGA, among others, which describe metals well but fail toaccount for the band gap of semiconductors. The empiricalmBJLDA potential is a response. An alternative way to dealwith this problem is the GW approximation (GWA). It is de-rived from many-body perturbation theory [30]. The form ofthe self-energy in the GWA is the same as in the HFA butthe Coulomb interaction is dynamically screened remedyingthe most serious deficiency of the HFA. The correspondingself-energy is therefore non-local and energy dependent. TheGreen function is obtained from a Dyson equation of the formG = G0 + G0ΣG whereG0 describes the direct propagationwithout the exchange and correlation interaction andΣ con-tains all possible exchange and correlation interactions withthe system that an electron can have in its propagation. TheGWA may be regarded as a generalization of the HFA butwith a dynamically screened Coulomb interaction. The non-local HFA is given by

Σx(r, r′) =(occ)∑

kn

ψ∗(r)ψ(r′)ν(r− r′) (8)

Whereν(r− r′) is the bare Coulomb interaction. The GWAcorresponds to replacing the bare Coulomb interactionν bya screened interaction W. In the language of perturbation the-ory this corresponds to

Σx(r, r′, ω) =i

2π

∫dω′G(r, r′, ω + ω′)W (r, r′, ω′) (9)

For details see Ref. 11. We will analyze the results obtainedfor the semiconductor gap value using this approximation inwhat follows.

In 2005, Rinke et al. [31] using the so calledOEPx(cLDA)+GW approximation obtained a reasonableagreement with experiment when calculating the band gapof a certain number of semiconductors. In 2007, Shishkin

et al. [32, 33] using a self-consistent GWA (GWA + DFT),and the self-consistent GW approximation with attractiveelectron-hole interaction, scGW(e-h) accounted quite wellfor the experimental band gap of several semiconductors.

Now we proceed to analyze some of the different offers inthe literature in what the calculation of the band gap of semi-conductors is concerned and compare their results amongthemselves and with experiment.

6. Analysis

In Table I, we present the results obtained for the band gapusing different methods and approximations. The first twocolumns refer to our calculation. The resulting values usingLDA and mBJLDA [20] as in version 2011 of Wien2k codeare presented. We have used as lattice parameter the averageof the values obtained from an LDA and a GGA optimiza-tion which, as we found [20], gives the best results for thegap as compared to experiment when the mBJLDA potentialis used. Next we report the values obtained with the hybridHISS and HSE06 functional [29], HSE [13, 14] and GWA.In the column denoted as GWA, we include the most precisepredictions for the gap reported in the literature using eitherthe self consistent GW (scGWA) [33–40] or the scGWA withattractive electron-hole interaction, scGW(e-h) [32].

In Fig. 1, the band gap value is given in the horizontalaxis. Each vertical line is drawn at the experimental low tem-perature band gap value for each of the 27 semiconductorsconsidered. The vertical axis represents the absolute percenterror, (|Error(%)|) calculated as shown at the bottom of Ta-ble I. The three data on each vertical line correspond to theresult obtained using the mBJLDA potential, the HSE methodand the GWA. So, for a particular semiconductor, the graphcompares directly the performance of each of the three bestmethods as found in this work. (see Table I).

As it is very well known [7] and as it appears in Ta-ble I, the LDA does not reproduce the experimental values ofthe band gap of semiconductors. Furthermore, The Wien2k(LDA) code produces for MgS and MgTe a band structurewhich shows a direct band gap in contradiction with experi-ment [20].

The results of the predictions obtained with the GWA arethe most accurate with an averaged error of 5.7%. The em-pirical mBJLDA potential produces results with an averagederror of 8.4%. Next, the errors obtained with the HSE poten-tial result in an averaged error of 10.2%, HSE06 (11.5%) andHISS (34.1%). The GWA, the mBJLDA potential, and HSEfunctional do better than the ones reported by Marquesetal. [28]. They get results with averaged errors 16.5%, 14.4%y 10.4% using the hybrid functional PBE0ε∞ , PBE0mix andHSE06mix, respectively. In this paper, the authors suggestthe possibility that the mixing parameter should be related tophysical variables. The performance of the GWA is highlyaccurate, 88% of the calculated results recorded here showless than 10% error. This is to be compared to the one ob-tained when using mBJLDA (74%), HSE (54%), HSE (42%),

Rev. Mex. Fis.59 (2013) 453–459

456 J. A. CAMARGO-MARTINEZ AND R. BAQUERO

TABLE I. We compare the results for the gap (Eg), in eV, that we obtained with the Wien2k(LDA) code and with mBJLDA potential, withthe hybrid functionals HISS, HSE06, HSE and with the GWA (see text). The crystal structure and percentage difference with respect to theexperiment is shown in parenthesis. The minus sign means that the calculation underestimates the experimental value. The experimental dataare from Refs. 41 to 49.

Gap

Solid ELDAg EmBJLDA

g EHISSg EHSE06

g EHSEg EGWA

g EExpt.g

The experimental band gap at Low temperature

C(A1) 4.16 (-24%) 4.95 (-9.7%) 6.11 (11.5%) 5.42 (-1.1%) 5.49 (0.2%) 5.6a (2.2%) 5.48

Si(A1) 0.45 (-62%) 1.17 (0.0%) 1.45 (23.9%) 1.22 (4.3%) 1.28 (9.4%) 1.24† (6.0%) 1.17

Ge(A1) 0.00 (-100%) 0.80 (8.1%) 1.08 (45.9%) 0.54 (-27.0%) 0.56 (-24.3%) 0.75a (1.4%) 0.74

MgO(B1) 5.00 (-36%) 7.22 (-7.1%) 7.87 (1.3%) 6.40 (-17.6%) 6.50(-16.3%) 7.7b (-0.9%) 7.77

AlAs(B3) 1.32 (-41%) 2.17 (-2.7%) 2.40 (7.6%) 2.16 (-3.1%) 2.24 (0.4%) 2.18c (-2.2%) 2.23

SiC(B3) 1.30 (-46%) 2.26 (-6.6%) 2.74 (13.2%) 2.32 (-4.1%) 2.39 (-1.2%) 2.53† (4.5%) 2.42

AlP(B3) 1.43 (-42%) 2.33 (-4.9%) 2.71 (10.6%) 2.44 (-0.4%) 2.52 (2.9%) 2.57† (4.9%) 2.45

GaN(B3) 1.90 (-46%) 2.94 (-10.9%) 4.05 (22.7%) 2.97 (-10.0%) 3.03 (-8.2%) 3.27† (-0.9%) 3.30

GaAs(B3) 0.47 (-69%) 1.56 (2.6%) 1.86 (22.4%) 1.18 (-22.4%) 1.21 (-20.4%) 1.52‡ (0.0%) 1.52

InP(B3) 0.45 (-68%) 1.52 (7.0%) 2.23 (57.0%) 1.61 (13.4%) 1.64 (15.5%) 1.44d (1.4%) 1.42

AlSb(B3) 1.14 (-32%) 1.80 (7.1%) 2.05 (22.0%) 1.85 (10.1%) 1.99 (18.5%) 1.64d (-2.4%) 1.68

GaSb(B3) 0.07 (-91%) 0.90 (9.8%) 1.31 (59.8%) 0.70 (-14.6%) 0.72 (-12.2%) 0.62d (-24.4%) 0.82

GaP(B3) 1.39 (-41%) 2.24 (-4.7%) 2.67 (13.6%) 2.42 (3.0%) 2.47 (5.1%) 2.55d (8.5%) 2.35

InAs(B3) 0.0 (-100%) 0.55 (31.0%) 0.93 (121.4%) 0.36 (-14.3%) 0.39 (-7.1%) 0.40d (-4.8%) 0.42

InSb(B3) 0.0 (-100%) 0.31 (29.2%) 0.80 (233.3%) 0.28 (16.7%) 0.29 (20.8%) 0.18d (-25%) 0.24

CdS(B3) 0.93 (-63%) 2.61 (5.2%) 2.72 (9.7%) 2.10 (-15.3%) 2.14 (-13.7%) 2.45e (-1.2%) 2.48

CdTe(B3) 0.49 (-69%) 1.67 (4.4%) 2.00 (25.0%) 1.49 (-6.9%) 1.52 (-5.0%) 1.76f (10.0%) 1.60

CdSe(B3) 0.38 (-79%) 1.87 (5.6%) 1.90 (7.3%) 1.36 (-23.2%) 1.39 (-21.5%) 2.01f (13.6%) 1.77

ZnS(B3) 2.08 (-45%) 3.70 (-2.9%) 4.12 (8.1%) 3.37 (-11.5%) 3.42 (-10.2%) 3.86‡ (1.3%) 3.81

ZnSe(B3) 1.19 (-58%) 2.74 (-2.8%) 2.93 (3.9%) 2.27 (-19.5%) 2.32 (-17.7%) 2.84f (0.7%) 2.82

ZnTe(B3) 1.20 (-50%) 2.38 (-0.4%) 2.77 (15.9%) 2.16 (-9.6%) 2.19 (-8.4%) 2.57f (7.5%) 2.39

MgS(B3) - 5.18 (-4.1%)* 5.17 (-4.3%) 4.48 (-17.0%) 4.78 (-11.5%) - 5.40

MgTe(B3) - 3.59 (-2.2%)* 3.91 (6.5 %) 3.49 (-4.9%) 3.74 (1.9%) - 3.67

GaN(B4) 2.06 (-41%) 3.13 (-10.6%) 4.23 (20.9%) 3.14 (-10.3%) 3.21 (-8.3%) 3.5g (0.0%) 3.50

InN(B4) 0.03 (-96%) 0.82 (15.5%) 1.51 (112.7%) 0.66 (-7.0%) 0.71 (0.0%) - 0.71

AlN(B4) 4.11 (-34%) 5.53 (-10.7%) 6.62 (6.9%) 5.50 (-11.2%) 6.45 (4.2%) 5.8g (-6.3%) 6.19

ZnO(B4) 0.76 (-78%) 2.76 (-19.8%) - - - 3.2† (-7.0%) 3.44

∆(%) 60.2% 8.4% 34.1% 11.5% 10.2% 5.7% -

The experimental band gap at room temperature

BP(B3) 1.15 (-43%) 1.83 (-8.5%) 2.43 (21.5%) 2.21 (10.5%) 2.16 (8.0%) - 2.00

BN(B3) 4.39 (-29%) 5.85 (-5.6%) 6.69 (7.90%) 5.90 (-4.8%) 5.99 (-3.4%) 7.14 (15.2%) 6.20

MgSe(B1) 1.71 (-31%) 2.89 (17.0%) 3.05 (23.5%) 2.58 (4.5%) 2.62 (6.1%) - 2.47

BaS(B1) 1.93 (-50%) 3.31 (-14.7%) 3.61 (-7.0%) 3.21 (-17.3%) 3.28 (-15.5%) 3.92 (1.0%)h 3.88

BaSe(B1) 1.74 (-51%) 2.87 (-19.8%) 3.14 (-12.3%) 2.80 (-21.8%) 2.87 (-19.8% - 3.58

BaTe(B1) 1.37 (-56%) 2.24 (-27.3%) 2.48 (-19.5%) 2.22 (-27.9%) 2.50 (-18.8%) - 3.08

BAs(B3) 1.23 (-16%) 1.72 (17.8%) 2.14 (46.6%) 1.89 (29.5%) 1.92 (31.5%) - 1.46

∗with mBJ(aLDA). †scGW(h-e) in Ref. 32.‡ scGW in Ref. 33.aRef. 34.bRef. 35.bRef. 36.dRef. 37.eRef. 38. fRef. 39.gRef. 40.hRef. 50.∆(%) is the

average of the absolute percent deviations. We calculate the percent deviation as follows Error(%) = (ETeo.g − EExpt

g ) ∗ 100/EExptg .

Rev. Mex. Fis.59 (2013) 453–459

THE BAND GAP PROBLEM: THE ACCURACY OF THE WIEN2K CODE CONFRONTED 457

FIGURE 1. The vertical axis is the absolute percent error (|Error(%)|). Each vertical line is drawn at the low-temperature experimental bandgap value for each semiconductor. It compares, the three best results (GWA, mBJLDA, and HSE) according to our analysis. The inset showsin more detail the semiconductors with a gap between 2.2 to 2.5 eV for clarity (see Table I and text).

HISS (35%). On the other hand, more than a 20% devia-tion from experiment occurs when using mBJLDA in 7% ofthe cases, and with the GWA in 8% of the semiconductorsstudied, which is to be compared with HSE (15%), HISS(46%). All together, the best results are obtained when us-ing the GWA; mBJLDA is next, but also HSE gives resultswith acceptable accuracy. A special case is the very-low-gapInSb. In this case none of the methods give less than a 20%error although it would be more reasonable to judge these re-sults from the absolute deviation in electron-volts rather thanfrom the percent deviation (see Table I). It is important forthe overall picture to stress that the GWA and our calcula-tions with the mBJLDA potential present deviations less than5% in 15 of the 27 semiconductors considered. When theHSE potential is considered, 12 of the 27 present less than a5% deviation.

One more observation. In the previous analysis we tookinto account only low temperature band gap data. In Ta-ble I, we also present some calculations for which we didnot found experimental reports at low temperature. Since thecalculations are done at 0K, room-temperature measurementsrequire extrapolation either using Varshni’s law or a quadraticfit or any other suitable method which, in any case, generatesan extra incertitude in the obtained 0K data. If we rather usethe high temperature data, the HSE potentials give a betteragreement with experiment.

Recently the mBJLDA potential has presented very goodresults in the study of complex systems. Is the case of theelectronic, and magnetic features of the metal-insulator tran-sition phase of VO2, which are well reproduced using themBJ potential [51]. This result does not reproduces correctlyusing the hybrid functional HSE [52].

7. Conclusions

The accurate calculation of the band gap of semiconductorsis a difficult task that has been the object of intense research

with the result of important progress during the last approx-imately ten years. As a continuation of our previos work(PRB) where we performed a detailed analysis of the per-formance of the recently published modified Becke-Johnsonpotential presented in this work our analysis of some differentsolutions and compare their results among them to pinpointthe actual accuracy of this empirical potential as componed toother methods. A group of 27 semiconductors (see Table I)for which we found low temperature data on the band gapvalue were considered. The results of the GWA, the Wien2kimplemented with the mBJLDA potential, and codes using ahybrid functional, HSE, and HISS were taken into consider-ation. The results reported by Marqueset al. [28] were foundto be less accurate than the ones of the GWA, the mBJLDApotential and the HSE functional (see text above for the pre-cise definition used here). The GWA was found to give, alltogether the best results. The mBJLDA potential producesresults slightly less accurate and HSE comes next. The twofirst methods give quite good results (prediction better than5% for 15 of the 27 semiconductors studied). In Fig. 1, wecompare the performance of the three best methods found inthis analysis for each of the 27 semiconductors separately. Itis important to stress the empirical character of the mBJLDApotential because it prevents the consistent definition of theoptimization procedure which contrasts with the sound basesof the GWA. Even with the several theoretical non-properlysolved issues, the mBJLDA potential gives rise to acceptablepredictions of the band gap value as compared to experiment.An extra factor to be taken into account is the computationalcost. If one would seek for precision without taking this fac-tor into account, the GWA is the method to use. If one wouldprefer to sacrifice a little the precision obtained against thesavings in computational cost, the mBJLDA potential seemsthe appropriate method. In conclusion, we can typify thestate of matters with respect to the calculation of the bandgap of semiconductors as follows. A quite precise methoddoes exist, the GWA approximation. It’s computational cost

Rev. Mex. Fis.59 (2013) 453–459

458 J. A. CAMARGO-MARTINEZ AND R. BAQUERO

is higher. A relatively quicker code, the Wien2k implementedwith the mBJLDA potential, gives somehow less accurate re-sults but quite acceptable at lower computational cost. Othermethods do exist but are less accurate. Very recently, the newapproximation announced in the Ref. [53] was implementedin the Wien2k 12.1 code for public use. The new hybrid func-tional YS-PBE0 is “equivalent” to the HSE one, according tothe authors. We will study this new functional in future work.

Acknowledgments

The authors acknowledge to the GENERAL COORDINA-TION OF INFORMATION AND COMMUNICATIONSTECHNOLOGIES (CGSTIC) at CINVESTAV for providingHPC resources on the Hybrid Cluster Supercomputer “Xi-uhcoatl”, that have contributed to the research results re-ported within this paper. J.A.C.M acknowledges the supportof Conacyt Mexico through a PhD scholarship.

1. W. Khon and L. J. Sham,Phys. Rev.140(1965) A1133.

2. J. P. Perdew and Y. Wang,Phys. Rev. B45 (1992) 13244.

3. J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha,Phys. Rev. Lett.82 (1999) 2544.

4. J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett.77(1996) 3865.

5. J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha,Phys. Rev. Lett.E78 (1997) 1396.

6. J. Tao, J. P. Perdew, V.N. Staroverov, and G. E. Scuseria,Phys.Rev. Lett.91 (2003) 146401.

7. J. P. Perdew,Int. J. Quantum Chem.30 (1986) 451.

8. G. A. Baraff and M. Schluter,Phys. Rev. B30 (1984) 3460.

9. V. I. Anisimov, J. Zaanen, and O. K. Andersen,Phys. Rev. B44(1991) 943.

10. L. Hedin,Phys. Rev.139(1965) A796.

11. F. Aryasetiawany, and O. Gunnarssonz,Rep. Prog. Phys.61(1998) 237.

12. W. G. Aulbur, L. Jonsson, and J. W. Wilkins,Solid State Phys.54 (2000) 1.

13. J. Heyd, G. E. Scuseria, and M. Ernzerhof,J. Chem. Phys.118(2003) 8207.

14. J. Heyd, J. E. Peralta, G.E. Scuseria, and R. L. Martin,J. Chem.Phys.123(2005) 174101.

15. J. Heyd, G. E. Scuseria, and M. Ernzerhof,J. Chem. Phys.124(2006) 219906.

16. T. M. Henderson, A. F. Izmaylov, G. E. Scuseria, and A. Savin,J. Chem. Phys.127(2007) 221103.

17. T. M. Henderson, A. F. Izmaylov, G. E. Scuseria, and A. SavinJ. Theor. Comput. Chem.4 (2008) 1254.

18. F. Tran and P. Blaha,Phys. Rev. Lett.102(2009) 226401.

19. P. Blaha, K. Schwars, G.K.H. Madsen, D. Kvasnicka, andJ. Luitz, WIEN2K:Full Potential-Linearized Augmented Planewaves and Local Orbital Programs for Calculating CrystalProperties, edited by K. Schwars, (Vienna University of Tech-nology, Austria,2001).

20. J. A. Camargo-Martınez, and R. Baquero,Phys. Rev. B86(2012) 195106.

21. A. D. Becke, and M. R. Roussel,Phys. Rev. A39 (1989) 3761.

22. A. D. Becke,J. Chem. Phys.98 (1993) 1372.

23. A. D. Becke,J. Chem. Phys.98 (1993) 5648.

24. M. Ernzerhof and J. P. Perdew,J. Chem. Phys.109(1998) 3313.

25. M. Ernzerhof, and G. E. Scuseria,J. Chem. Phys.110 (1999)5029.

26. C. Adamo, and V. Barone,J. Chem. Phys.110(1999) 6158.

27. J. P. Perdew, M. Ernzerhof, and K. Burke,J. Chem. Phys.105(1996) 9982.

28. M. A. L. Marques, J. Vidal, M. J. T. Oliveira, L. Reining, andS. Botti,Phys. Rev. B83 (2011) 035119.

29. M. J. Lucero, T. M. Henderson, and G. E. Scuseria,J. Phys.:Condens. Matter24 (2003) 145504.

30. A. L. Fetter and J. D. Walecka,Quantum Theory of Many-Particle Systems.(Courier Dover Publications, 2003).

31. P. Rinkeet al., New J. Phys.7 (2005) 126.

32. M. Shishkin, M. Marsman, and G. Kresse,Phys. Rev. Lett.99(2007) 246403.

33. M. Shishkin, and G. Kresse,Phys. Rev. B75 (2007) 235102.

34. M.S. Hybertsen, and S. G. Louie,Phys. Rev. B34 (1986) 5390.

35. U. Schonberger, and F. Aryasetiawan,Phys. Rev. B52 (1995)8788.

36. R. W. Godby, M. Schluter, and L. J. Sham,Phys. Rev. B37(1988) 10159.

37. X. Zhu and S. G. Louie,1Phys. Rev. B43 (1991) 14142.

38. M. Rohlfing, P. Kruger, and J. Pollmann,Phys. Rev. Lett.75(1995) 3489.

39. O. Zakharov, A. Rubio, X. Blase, M. L. Cohen, and S. G. Louie,Phys. Rev. B50 (1994) 10780.

40. A. Rubio, J. L. Corkill, M. L. Cohen, E. L. Shirley, and S. G.Louie,Phys. Rev. B48 (1993) 11810.

41. S. Adachi,Handbook on Physical Properties of Semiconduc-tors, Vols. I, II and III.(Kluwer Academic Publishers, 2004).

42. O. Madelung,Data in Sciene and Technology, SemiconductorsGroup IV Elements and II-V Compounds. (Ed. Springer-Verlag,1991).

Rev. Mex. Fis.59 (2013) 453–459

THE BAND GAP PROBLEM: THE ACCURACY OF THE WIEN2K CODE CONFRONTED 459

43. O. Madelung,Semiconductors: Data Handbook CD-ROM.(Ed. Springer-Verlag, 2003).

44. D. Wolverson, D. M. Bird, C. Bradford, K. A. Prior, and B. C.Cavenett,Phys. Rev. B, 64 (2001) 113203.

45. K. Watanabeet al., J. Appl. Phys.81 (1997) 451.

46. M. Feneberg, J. Daubler, K. Thonke, R. Sauer, P. Schley, andR. Goldhahn,Phys. Rev. B, 77 (2008) 245207.

47. H. Morkoc,Handbook of Nitride Semiconductors and Devices,Ed. Wiley-VCH (2008).

48. D. M. Roessler and W. C. Walker,J. Phys. Chem. Solids28(1967) 1507.

49. G. Ramırez-Flores, H. Navarro-Contreras, A. Lastras-Martınez,R. C. Powell and J. E. Greene,Phys. Rev. B50 (1994) 8433.

50. L. Tie-Yu, C. De-Chan and H. Mei-Chun,Chinese Phys. Lett.23 (2005) 943.

51. R. Grau-Crespo, H. Wang and U. Schwingenschlogl,Phys. Rev.B 86 (2012) 081101(R).

52. Z. Zhu and U. Schwingenschlogl,Phys. Rev. B86 (2012)075149.

53. F. Tran and P. Blaha,Phys. Rev. B83 (2011) 235118.

Rev. Mex. Fis.59 (2013) 453–459